A Fractional Order Recovery SIR Model from a Stochastic Process

Area of Science:

• Epidemiology
• Mathematical Biology
• Stochastic Processes

Background:

• Epidemiological models often use fractional derivatives without clear justification.
• The relevance of ad hoc fractional models to real-world disease dynamics is uncertain.
• A need exists for a rigorous derivation of fractional epidemiological models.

Purpose of the Study:

• To develop a fractional SIR (Susceptible-Infected-Recovered) model for epidemics from an underlying stochastic process.
• To investigate the conditions under which fractional differential operators naturally arise in epidemiological models.
• To provide a mathematically sound basis for fractional epidemiological modeling, particularly for chronic diseases.

Main Methods:

• Developed a SIR model incorporating vital dynamics from a stochastic process.
• Demonstrated that fractional differential operators emerge when recovery times are power-law distributed.
• Extended the stochastic derivation to discrete time, enabling stable numerical solutions.
• Analyzed model consistency with existing SIR models (Kermack-McKendrick, Hethcote-Tudor).

Main Results:

• Fractional derivatives naturally arise in SIR models with power-law distributed recovery times, modeling chronic disease.
• The fractional order recovery model is consistent with established SIR models.
• Simulations confirmed convergence to equilibrium states.
• The number of infected individuals at endemic equilibrium increases as the fractional order approaches zero.

Conclusions:

• Fractional calculus provides a natural framework for modeling epidemics with power-law recovery, relevant to chronic diseases.
• The stochastic derivation offers a robust foundation for fractional epidemiological models.
• Model behavior indicates that a lower fractional order (closer to zero) leads to higher endemic infection levels.